1. Drawing a graph

2.
Graph Theory and Complex Networks by Maarten van Steen

3. Graph Theory and Complex Networks by Maarten van Steen

4. What is a planar embedding?
     drp.math.umd.edu/Project-Slides/Characteristics of Planar Graphs.pptx
What is a planar embedding? K4

5. Kuratowski’s Theorem (1930)
A graph is planar if and only if it does not contain a subdivision of K5 or K3,3.

6. Kuratowski Subgraphs What is a subdivision? K5 K3,3
What is a subdivision?
Kuratowski Subgraphs

7. For a planar connected graph |V| - |E| + |F| = 2
Euler characteristic (simple form):
= number of vertices – number of edges + number of faces
Or in short-hand,
= |V| - |E| + |F|
where V = set of vertices
E = set of edges
F = set of faces = set of regions
& the notation |X| = the number of elements in the set X.
For a planar connected graph |V| - |E| + |F| = 2

8. Defn: A tree is a connected graph that does not contain a cycle.
A forest is a graph whose components are trees.
Lemma 2.1: Any tree with n vertices has n-1 edges.
χ = 8 – = χ = 8 – = χ = 8 – 9 + 3= 2

9. = |V| – |E| + |F|
= 1 – = 2

10. = |V| – |E| + |F|
= 2 – = 2

11. = |V| – |E| + |F|
= 3 – = 2

12. = |V| – |E| + |F|
= 4 – = 2

13. = |V| – |E| + |F|
= 5 – = 2

14. = |V| – |E| + |F|
= 8 – = 2

15. = |V| – |E| + |F|
= 8 – = 2
Not a tree.

16. = 8 – 9 + 3 = 2 Not a tree. = |V| – |E| + |F|
= 8 – = 2
For the brave of heart, consider graphs drawn on other surfaces such as a torus or Klein bottle. For fun, see or
Not a tree.

17. Euler’s fomula: For a planar connected graph |V| - |E| + |F| = 2
where V = set of vertices, E = set of edges, F = set of faces = set of
regions
Defn: A tree (or acyclic graph) is a connected graph that does not contain a cycle.
A forest is a graph whose components are trees.
Lemma 2.1: Any tree with n vertices has n-1 edges.
Thm 2.9: For any connected planar graph with |V| ≥ 2,
|E| ≤ 3|V| - 6
Cor 2.4: K5 is nonplanar.
Thm 2.10: K3,3 is nonplanar.
Cor: A graph is planar if and only if it does not contain a subdivision of K5 or K3,3.